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FINDING THE LIMIT OF INCOMPLETENESS II

YONG CHENG

Abstract. This work is motivated from finding the limit of theapplicability of Godel’s first incompleteness theorem (G1): can wefind a minimal theory in some sense for which G1 holds? Theanswer of this question depends on our definition of minimality.We first show that the Turing degree structure of r.e. theories forwhich G1 holds is as complex as the structure of r.e. Turing degrees.Then we examine the interpretation degree structure of r.e. theoriesweaker than the theory R for which G1 holds, and answer someopen questions about this structure in the literature.

1. Preliminaries

Robinson Arithmetic Q and the theory R are both introduced in[15] by Tarski, Mostowski and Robinson as base axiomatic theories forinvestigating incompleteness and undecidability.Definition 1.1 (Robinson Arithmetic Q). Robinson Arithmetic Q isdefined in the language 0,S,+,× with the following axioms:

Q1: ∀x∀y(Sx = Sy → x = y);Q2: ∀x(Sx 6= 0);Q3: ∀x(x 6= 0 → ∃y(x = Sy));Q4: ∀x∀y(x+ 0 = x);Q5: ∀x∀y(x+ Sy = S(x+ y));Q6: ∀x(x× 0 = 0);Q7: ∀x∀y(x× Sy = x× y + x).

Definition 1.2 (The theory R). Let R be the theory consisting ofschemes Ax1−Ax5 in the language 0,S,+,×,≤.1

Ax1: m+ n = m+ n;Ax2: m× n = m× n;Ax3: m 6= n if m 6= n;Ax4: ∀x(x ≤ n→ x = 0 ∨ · · · ∨ x = n);Ax5: ∀x(x ≤ n ∨ n ≤ x).

2010 Mathematics Subject Classification. 03F40, 03F30, 03D35.Key words and phrases. Godel’s first incompleteness theorem, Essentially unde-

cidable, Interpretation, The theory R.We would like to thank Albert Visser for the email communications from which

some results in this paper are inspired, and for his contributions in this field thatthis paper is based on as we have specified in the paper. We also thank FedorPakhomov for the email communications of his work. To be updated ...

1For any n ∈ ω, we define the term n as follows: 0 = 0, and n+ 1 = Sn.0

FINDING THE LIMIT OF INCOMPLETENESS II 1

The theory R contains all key properties of arithmetic for the proofof Godel’s first incompleteness theorem (G1). Unlike Q, the theory R

is not finitely axiomatizable.

Definition 1.3 (Translations and interpretations).

• Let T be a theory in a language L(T ), and S a theory in alanguage L(S). In its simplest form, a translation I of languageL(T ) into language L(S) is specified by the following:

– an L(S)-formula δI(x) denoting the domain of I;– for each relation symbol R of L(T ), as well as the equalityrelation =, an L(S)-formula RI of the same arity;

– for each function symbol F of L(T ) of arity k, an L(S)-formula FI of arity k + 1.

• If φ is an L(T )-formula, its I-translation φI is an L(S)-formulaconstructed as follows: we rewrite the formula in an equivalentway so that function symbols only occur in atomic subformulasof the form F (x) = y, where xi, y are variables; then we replaceeach such atomic formula with FI(x, y), we replace each atomicformula of the form R(x) with RI(x), and we restrict all quan-tifiers and free variables to objects satisfying δI . We take careto rename bound variables to avoid variable capture during theprocess.

• A translation I of L(T ) into L(S) is an interpretation of T inS if S proves the following:

– for each function symbol F of L(T ) of arity k, the formulaexpressing that FI is total on δI :

∀x0, · · · ∀xk−1(δI(x0)∧· · ·∧δI(xk−1) → ∃y(δI(y)∧FI(x0, · · · , xk−1, y)));

– the I-translations of all axioms of T , and axioms of equal-ity.

The simplified picture of translations and interpretations above ac-tually describes only one-dimensional, parameter-free, and one-piecetranslations. For precise definitions of a multi-dimensional interpreta-tion, an interpretation with parameters, and a piece-wise interpretation,we refer to [19] for more details.Definition 1.4 (Interpretations II).

• A theory T is interpretable in a theory S if there exists aninterpretation of T in S.

• Given theories S and T , let ‘ST ’ denote that S is interpretablein T (or T interprets S); let ‘S T ’ denote that T interpretsS but S does not interpret T ; we say S and T are mutuallyinterpretable, denoted by S ≡I T , if S T and T S.

• We say that the theory S is weaker than the theory T w.r.t. in-terpretation if S T .

2 YONG CHENG

The notion of interpretation provides us a method to compare differ-ent theories in different languages. If T is interpretable in S, then allsentences provable (refutable) in T are mapped, by the interpretationfunction, to sentences provable (refutable) in S.2 The interpretationrelation among first order theories () is reflexive and transitive. Theequivalence classes of theories, under the equivalence relation ≡I , arecalled the interpretation degrees.3

In this paper, we work with first-order theories with finite signa-ture, and always assume the arithmetization of the base theory. Underarithmetization, we equate a set of sentences with the set of Godel’snumbers of sentences in it.Definition 1.5. Let ⊑ be a binary relation on r.e. theories.

(1) For r.e. theories S and T , define that S T iff S ⊑ T and T ⊑ Sdoes not hold.

(2) We say S is a minimal theory w.r.t. the relation ⊑ if there is notheory T such that T S.

(3) We say S is a maximal theory w.r.t. the relation ⊑ if there is notheory T such that S T .

Definition 1.6 (Folklore).

(1) Given two arithmetic theories U and V , U ≤T V denotes that thetheory U is Turing reducible to the theory V , and U <T V denotesthat U ≤T V but V T U .

(2) We say that the theory S is weaker than the theory V w.r.t. Turingreducibility if S <T V .

(3) We say a set A separates B and C if B ⊆ A and A ∩ C = ∅.(4) We say 〈S, T 〉 is a recursively inseparable pair if S and T are disjoint

r.e. subsets of ω, and there is no recursive set X ⊆ ω such that Xseparates S and T .

(5) Let 〈We : e ∈ ω〉 be the list of all r.e. sets, and 〈ϕe : e ∈ ω〉 be thelist of all Turing programs.

(6) A theory T is essentially undecidable if any recursively axiomatiz-able consistent extension of T in the same language is undecidable.

(7) A theory T is essentially incomplete if any recursively axiomatiz-able consistent extension of T in the same language is incomplete.4

Note that x ∈ We if and only if for some y, the e-th Turing programwith input x yields an output in less than y steps. We assume thatsuch y is unique if it exists.

2If theories S and T are mutually interpretable, then T and S are equally strongw.r.t. interpretation.

3In this paper, we only consider countable theories. There are 2ω countabletheories, and 2ω associated interpretation degrees.

4The theory of completeness/incompleteness is closely related to the theory ofdecidability/undecidability (see [15]).

FINDING THE LIMIT OF INCOMPLETENESS II 3

A lattice can be considered as algebraic structures with a signatureconsisting of two binary operations ∧ and ∨.

Definition 1.7. The theory of lattice consists of the following axioms:

Commutative laws: ∀a∀b(a ∨ b = b ∨ a); ∀a∀b(a ∧ b = b ∧ a).Associative laws: ∀a∀b∀c(a ∨ (b ∨ c) = (a ∨ b) ∨ c); ∀a∀b∀c(a ∧(b ∧ c) = (a ∧ b) ∧ c).

Absorption laws: ∀a∀b(a ∨ (a ∧ b) = a); ∀a∀b(a ∧ (a ∨ b) = a).Distributive: ∀x∀y∀z(x∨(y∧z) = (x∨y)∧(x∨z)); ∀x∀y∀z(x∧(y ∨ z) = (x ∧ y) ∨ (x ∧ z)).

The following theorem provides us with a method for proving theessentially undecidability of a theory via interpretation.

Theorem 1.8 (Theorem 7, Corollary 2, [15]). Let T1 and T2 be twoconsistent theories with finite signature such that T2 is interpretable inT1. If T2 is essentially undecidable, then T1 is essentially undecidable.5

Lemma 1.9 (The fixed point lemma, Folklore). Let T be a consis-tent r.e. extension of Q. For any formula φ(x) with exactly one freevariable, there exists a sentence θ such that T ⊢ θ ↔ φ(pθq).

2. Introduction

Let T be a consistent r.e. theory. To generalize G1 to weaker theoriesthan PA w.r.t. interpretation, we introduce the notion “G1 holds forT”.

Definition 2.1 (Cheng, [1]). We say that G1 holds for a r.e. theory Tif any consistent r.e. theory that interprets T is incomplete.

Proposition 2.2 (Cheng, [1]). G1 holds for T iff T is essentially in-complete iff T is essentially undecidable.

It is well known that G1 holds for Robinson Arithmetic Q and thetheory R (see [15]). In fact, G1 holds for many theories weaker thanPA w.r.t. interpretation. In summary, we have the following picture:6

• Q IΣ0 + exp IΣ1 IΣ2 · · · IΣn · · · PA, and G1

holds for them.• The theories Q, IΣ0, IΣ0 +Ω1, · · · , IΣ0 +Ωn, · · · , BΣ1, BΣ1 +Ω1, · · · , BΣ1 + Ωn, · · · are all mutually interpretable, and G1

holds for them.• Theories PA−,Q+,Q−,TC,AS,S1

2and Q are all mutually in-

terpretable, and G1 holds for them.• RQ EAPRAPA, and G1 holds for them.

5For theories with infinite signature, this theorem does not hold.6For the definition of these weak theories, we refer to [1].

4 YONG CHENG

This paper is motivated from finding the limit of the applicability ofG1: can we find a minimal theory in some sense for which G1 holds?The answer of this question depends on our definition of minimality.

If we define minimality as having the minimal number of axiom, thenany finitely axiomatized essentially undecidable theory (e.g., RobinsonArithmetic Q) is a minimal r.e. theory for which G1 holds. For theorieswhich is not finitely axiomatized, if we define minimality as having theminimal number of axiom schemes, then the following theory VS isa minimal r.e. theory for which G1 holds since it has only one axiomscheme and is essentially undecidable. The Vaught set theory VS,originally introduced by Vaught [16], is axiomatized by the schema

(Vn) ∀x0, · · · , ∀xn−1∃y∀t(t ∈ y ↔∨

i<n

t = xi)

for all n ∈ ω, asserting that xi : i < n exists.When we talk about minimality, we should specify the degree struc-

ture involved. In [1], we examine the following two degree structuresthat are respectively induced from Turing reducibility and interpreta-tion: 〈D,≤T 〉 and 〈D,〉.Definition 2.3 (Cheng, [1]).

(1) Let D = S : S <T R, and G1 holds for the r.e. theory S.(2) Let D = S : S R and G1 holds for the r.e. theory S.

In [1], we show that there is no minimal r.e. theory w.r.t. Turingreducibility for which G1 holds, and prove some results about the struc-ture of 〈D,〉. In this paper, we prove more facts about 〈D,≤T 〉, andanswer open questions about the structure 〈D,〉 in [1]. Moreover,we prove in Theorem ?? that the index set of r.e. theories for whichG1 holds is Π0

3-complete. As a corollary, we show that for some degreestructures satisfying the conditions in Theorem ??, there is no minimalr.e. theory for which G1 holds.

3. The structure 〈D,≤T 〉

In this section, we examine the Turing degree structure of r.e. theo-ries below R for which G1 holds.

Hanf shows that there is a finitely axiomatizable theory in each re-cursively enumerable tt-degree (see [13]). Feferman shows in [3] thatif A is any recursively enumerable set, then there is a recursively ax-iomatizable theory T having the same Turing degree as A. In [14],Shoenfield improves Feferman’s result and shows that if A is not re-cursive, then there is an essentially undecidable theory with the sameTuring degree. To make readers to have a good sense of Shoenfield’stheorem, we give a proof of it which is a reconstruction of Shoenfield’sproof in [14] with more details.

FINDING THE LIMIT OF INCOMPLETENESS II 5

Theorem 3.1 (Shoenfield, [14]). If A is recursively enumerable andnot recursive, there is a recursively inseparable pair 〈B,C〉 such thatA, B and C have the same Turing degree.

Theorem 3.2 (Shoenfield, [14]). Let A be recursively enumerable andnot recursive. Then there is a consistent axiomatizable theory T hav-ing one non-logical symbol which is essentially undecidable and has thesame Turing degree as A.

Proof. Pick a recursively inseparable pair 〈B,C〉 as in Theorem 3.1such that A, B and C have the same Turing degree. The theory Twe define has only one non-logical symbol: a binary relation symbolR. Let Φn be the statement that there is an equivalence class of Rconsisting of n elements. The theory T contains the following axioms:

• axioms asserting that R is an equivalence relation;• Φn for all n ∈ B;• ¬Φn for all n ∈ C;• for each n we adopt an axiom asserting there is at most oneequivalence class of R having n elements.

Note that T is consistent and axiomatizable. Since Φn is provable iffn ∈ B, and ¬Φn is provable iff n ∈ C, we have B and C are recursivein T .

Disjunctions of conjunctions whose terms are Φn or ¬Φn for somen ∈ ω, are called a disjunctive normal form of 〈Φn : n ∈ ω〉.

Lemma 3.3 (Janiczak, Lemma 2 in [6]). Any sentence φ of the theoryT is equivalent to a disjunctive normal form of 〈Φn : n ∈ ω〉, andthis disjunctive normal form can be found explicitly once φ is explicitlygiven.7

By Lemma 3.3, every sentence φ of T is equivalent to a disjunctivenormal form of 〈Φn : n ∈ ω〉, and this disjunctive normal form can becalculated from φ. It follows that T is recursive in (B,C). Hence, Thas the same Turing degree as A. Finally, by a standard argument, wecan show that T is essentially undecidable (see [1]).

Corollary 3.4. The structure 〈D,≤T 〉 is as complex as the Turingdegree structure of r.e. sets.

However, Shoenfield’s theory T in Theorem 3.2 is not unique as canbe seen from the following theorem which improves Theorem 3.1.

Theorem 3.5. For any non-recursive r.e. set A, there is a countablesequence of r.e. sets 〈Bn : n ∈ ω〉 such that (Bn, Bm) is a recursivelyinseparable pair for m 6= n and each Bn has same Turing degree as A.

7This is a reformulation of Janiczak’s Lemma 2 in [6] in the context of the theoryT . Janiczak’s Lemma is proved by means of a method known as the elimination ofquantifiers.

6 YONG CHENG

Proof. Suppose A is a non-recursive r.e. set. We construct a sequenceof disjoint r.e. sets 〈Bn : n ∈ ω〉 such that for any m 6= n, (Bn, Bm) isa recursively inseparable pair and Bn ≡T Bm ≡T A.

Let f be the recursive function that enumerates A without rep-etitions. Define a partial function g as follows: g(〈x, y〉) = n iff∃s∃t(f(s) = x, t < s and the program ϕs with input 〈x, y〉 yields nas output in ≤ t steps).

By the projection theorem, g is partial recursive. Let Bn = x :g(x) = n. For any n 6= m, we show that Bn and Bm have the sameTuring degree as A and (Bn, Bm) is a recursively inseparable pair. Notethat Bn and Bm are disjoint r.e. sets.

Claim. Bn ≤T A.

Proof. We test 〈x, y〉 ∈ Bn as follows. If x /∈ A, then 〈x, y〉 /∈ Bn. Ifx ∈ A, let s = f−1(x). We can decide whether the program ϕy withinput 〈x, y〉 yields n in < s steps. If yes, then 〈x, y〉 ∈ Bn; if no, then〈x, y〉 /∈ Bn.

Claim. A ≤T Bn.

Proof. We test x ∈ A as follows. Suppose the index of the program withconstant output value n is e0, i.e. ϕe0(x) = n for any x. If 〈x, e0〉 ∈ Bn,then x ∈ A. Suppose 〈x, e0〉 /∈ Bn, let w be the number of computationsteps such that ϕe0(〈x, e0〉) = n. Decide whether f(s) = x and s ≤ wfor some s. If yes, then x ∈ A; if no, then x /∈ A.

Similarly, we can show that Bm ≤T A and A ≤T Bm.

Claim. There is no recursive set X that separates Bn and Bm.

Proof. Suppose not. Then we can find a recursive function h such thatran(h) = m,n, h(Bn) = m and h(Bm) = n. Let h = ϕe1 .

Claim. If x ∈ A, then the program ϕe1 with input 〈x, e1〉 yields outputin at least s steps where s = f−1(x).

Proof. Suppose not. Then the program ϕe1 with input 〈x, e1〉 halts inless than s = f−1(x) steps. Then by definition, we have ϕe1(〈x, e1〉) =n⇔ ϕe1(〈x, e1〉) = m, which leads to a contradiction.

Let l be the recursive function such that l(x) = the number of stepsto compute the value of 〈x, e1〉 in the program ϕe1. Then we have:

x ∈ A⇔ ∃s(s ≤ l(x) ∧ f(s) = x).

Thus, A is recursive which leads to a contradiction.

Thus, for any n 6= m, (Bn, Bm) is a recursively inseparable pair andBn ≡T Bm ≡T A.

FINDING THE LIMIT OF INCOMPLETENESS II 7

As a corollary, for any no-recursive r.e. degree a, there are countablymany essentially undecidable r.e. theories with the same Turing degreea. Since there are only countably many r.e. theories, this is the bestresult we can have.

Now, we list some key results about the Turing degree structure ofr.e. sets.Fact 3.6 (Folklore, many authors).

• The r.e. degrees are dense: for any r.e. sets A <T B, there is ar.e. set C such that A <T C <T B.

• No r.e. degree is minimal.• For any r.e. set 0 <T C <T 0′, there exists an r.e. set Asuch that A is incomparable with C w.r.t. Turing degree. Fur-thermore, an index for A can be found uniformly from one forC.

• Given r.e. sets A <T B, there is an infinite r.e. sequence of r.e.sets Cn such that A <T Cn <T B and Cn’s are incomparablew.r.t. Turing degree.

• If a and b are r.e. degrees such that a < b, then any count-ably partially ordered set can be embedded in the r.e. degreesbetween a and b.

Corollary 3.7.

• 〈D,≤T 〉 is dense: for theories A,B ∈ D such that A <T B,there is a theory C ∈ D such that A <T C <T B.

• For any theory A ∈ D, there exists a theory B ∈ D such thatB is incomparable with A under ≤T . As a corollary, for anytheory A ∈ D, there exists a theory B ∈ D such that A <T B.

• 〈D,≤T 〉 has no minimal element, and has no maximal element.• Given theories A,B ∈ D such that A <T B, there is an infiniter.e. sequence of theories Cn ∈ D such that A <T Cn <T B andCn’s are incomparable w.r.t. Turing degree.

• Given theories A,B ∈ D such that A <T B, any countablypartially ordered set can be embedded in 〈D,≤T 〉 between A andB.

Thus, 〈D,≤T 〉 is a dense distributive lattice without endpoints.

4. Some interpretation degree structures

In this section, before we examine the structure of 〈D,〉, we firstreview some interpretation degree structures we know in the literature.Let 〈DPA,〉 denote the interpretation degree structure of consistentr.e. extensions of PA. For consistent r.e. extensions of PA, we havesome equivalent characterizations of the notion of interpretation.

Fact 4.1 (Folklore, [8]). Suppose theories S, T are consistent r.e. ex-tensions of PA. Then the following are equivalent:

8 YONG CHENG

(1) S T ;(2) T ⊢ Con(S k) for any k ∈ ω;8

(3) (S k) T for any k ∈ ω;(4) For any Π0

1 sentence φ, if S ⊢ φ, then T ⊢ φ.(5) For every model M of T , there is a model N of S such that M is

isomorphic to an initial segment of N .

The structure 〈DPA,〉 is well known in the literature. In fact,〈DPA,, ↓, ↑〉 is a dense distributive lattice.9 For more properties of〈DPA,〉, we refer to [8].

Now, we examine the interpretation degree structure of general r.e.theories, not restricting to theories interpreting PA.

Definition 4.2 (Folklore). We introduce two natural operators on r.e.theories.

• The supremum A⊗ B is defined as follows: A⊗ B is a theoryin the disjoint sum of the signatures of A and B plus two newpredicates P0 and P1. We have axioms that say that P0 and P1

form a partition of the domain and the axioms of A relativizedto P0 and the axioms of B relativized to P1.

• The infimum A⊕ B is defined as follows: A⊕B is a theory inthe disjoint sum of the signatures of A and B plus a fresh 0-arypredicate symbol P . The theory is axiomatized by all P → ϕ,where ϕ is an axiom of A plus ¬P → ψ, where ψ is an axiomof B.

Note that the interpretation degree structure of r.e. theories withthe operators ⊕ and ⊗ is a distributive lattice.

The interpretation degree structure of all finitely axiomatized theo-ries, denoted by 〈Dfinite,〉, is studied by Harvey Friedman in [Fri07].Note that there are only ω many interpretation degrees of finitely ax-iomatized theories. The structure 〈Dfinite,〉 forms a (reflexive) partialordering with a minimum element ⊤ and a maximum element ⊥ where⊤ is the equivalence class of all sentences with a finite model, and ⊥ isthe equivalence class of all sentences with no models.Theorem 4.3 (Harvey Friedman, [4]).

(1) The structure 〈Dfinite,,⊕,⊗,⊥,⊤〉 forms a distributive lattice.(2) For any a ∈ Dfinite such that a ⊥, there exists b ∈ Dfinite such

that a b⊥.(3) The structure 〈Dfinite,〉 is dense, i.e., a b→ (∃c)(a c b).(4) For any a, b ∈ Dfinite, if ab, then there exists an infinite sequence

cn such that a cn b for each n and cn’s are incomparable w.r.t.interpretation.

8Con(S) is the canonical arithmetic formula which expresses the consistency ofthe theory S saying that 0 6= 0 is not provable in S.

9For consistent r.e. extensions A and B of PA, A ↓ B = PA + Con(A k) ∨ Con(B k) : k ∈ ω and A ↑ B = PA+ Con(A k) ∧ Con(B k) : k ∈ ω.

FINDING THE LIMIT OF INCOMPLETENESS II 9

Thus, 〈Dfinite,,⊕,⊗,⊥,⊤〉 is a dense distributive lattice withoutendpoints. Especially, there is no minimal finitely axiomatized theoryw.r.t. interpretation.

The following theorem shows that above any finitely axiomatizablesub-theory ofPA, there are continuum many incomparable sub-theoriesof PA w.r.t. interpretation.

Theorem 4.4 (Montague, [9]). Let T be any finitely axiomatizablesubtheory of PA. Then there is a set C of cardinality 2ω such that (i)every member of C is a sub-theory of PA and an extension of T , (ii)any two distinct elements of C are incomparable w.r.t. interpretation.

The structure 〈Dfinite,〉 bears a rough resemblance to the Turingdegree structure of r.e. sets which is very well studied from recursiontheory, and is very complicated. It is not fully clear how complicated〈Dfinite,〉 is, and how these two structures are related. Moreover,there are many similarities between the structure 〈Dfinite,〉 and the

structure 〈D,≤T 〉; an interesting question is: what is the differencebetween these two structures?

Definition 4.5 ([17]).

(1) An interpretation is direct when it is un-relativized and identitypreserving.

(2) A theory is sequential if it directly interprets the theory AdjunctiveSet Theory (AS).10

(3) We say two sentences have the same derivability degree iff they areprovably equivalent over EA.

Let 〈DSeq,〉 denote the interpretation degree structure of finitelyaxiomatized sequential theories. The following theorem gives us a nicecharacterization of the structure 〈DSeq,〉.

Theorem 4.6 (Visser, [17]). The structure 〈DSeq,〉 is recursivelyequivalent to the degrees of derivability of Π0

1-sentences over EA.

5. The structure 〈D,〉

In this section, we examine the interpretation degree structure ofr.e. theories for which G1 holds. The interpretation degree structure ofr.e. theories interpreting PA is well known. However, the interpretationdegree structure of r.e. theories weaker than the theory R is much morecomplex and we know few about it.

Now, we examine the interpretation degree structure of r.e. theoriesweaker than R for which G1 holds, and answer open questions aboutthe structure 〈D,〉 in [1].

10The theory AS has only one binary relation symbol ‘∈’, and the followingaxioms: (1) ∃x∀y(y /∈ x); (2) ∀x∀y∃z∀u(u ∈ z ↔ (u = x ∨ u = y)).

10 YONG CHENG

Definition 5.1 ([1]). Let 〈A,B〉 be a recursively inseparable pair. Con-sider the following r.e. theory U〈A,B〉 with the signature 0,S,P whereP is a unary relation symbol, and n = Sn0 for n ∈ N:

(1) m 6= n if m 6= n;(2) P(n) if n ∈ A;(3) ¬P(n) if n ∈ B.

Theorem 5.2 (Cheng, [1]). For any recursively inseparable pair 〈A,B〉,G1 holds for U〈A,B〉 and U〈A,B〉 R.

Since there are countably many recursively inseparable pairs, thereare countably many elements of D.

Definition 5.3 (Visser). We say a r.e. theory U is Turing persistentif for any consistent r.e. theory V , if U ⊆ V , then U ≤T V .

There is no direct relation between the notion of interpretation andthe notion of Turing reducibility. Given r.e. theories U and V , U Vdoes not imply U ≤T V , and U ≤T V does not imply UV . The notionof “Turing persistent” establishes the relationship between U V andU ≤T V . Note that if U is Turing persistent, then for any r.e. theoryV , if U V , then U ≤T V . Many essentially undecidable theorieswe know are Turing persistent. A natural question is: can we find anessentially undecidable theory which is not Turing persistent? Now,we give some examples of Turing persistent theories.

Proposition 5.4. For any consistent r.e. theory T , if all recursivefunctions are representable in T , then T is Turing persistent.11

Proof. This follows from the fact that T has Turing degree 0′ since anyr.e. set is representable in T .

As a corollary, R is Turing persistent.

Theorem 5.5 (Essentially due to Shoenfield). For any r.e. set A, thereare disjoint r.e. sets B and C with B,C ≤T A such that for any r.e. Dwhich separates B and C, we have A ≤T D.12

Proof. Suppose that A = We, i.e. x ∈ A if and only if for some y, thee-th Turing program with input x yields an output in < y steps.

• Define x ∈ B iff for some y, the e-th Turing program with input(x)0 yields an output in < y steps and for all z ≤ y, the (x)1-thTuring program with input x does not yield an output in < zsteps.

11Our notion of representability is standard. We refer to [10] for definitions.12The proof of this theorem is based on Shoenfield’s construction in [14]. Albert

Visser also discovered this form in his note on Shoenfield’s theorem.

FINDING THE LIMIT OF INCOMPLETENESS II 11

• Define x ∈ C iff for some y, the e-th Turing program withinput (x)0 yields an output in < y steps and for some z ≤ y,the (x)1-th Turing program with input x yields an output in< z steps.

Note that B and C are disjoint r.e. sets. If (x)0 /∈ A, then x /∈ Band x /∈ C; if (x)0 ∈ A, then we can decide either x ∈ B or x ∈ C.Thus, we have B,C ≤T A.

Suppose D is a r.e. set with index d, and B ⊆ D and D ∩ C = ∅.We show that A ≤T D.Claim. x ∈ A if and only if for some z, the d-th Turing program withinput 〈x, d〉 yields an output in < z steps and for some y < z, the e-thTuring program with input x yields an output in < y steps.

Proof. The right-to-left direction is obvious. Suppose x ∈ A. Theneither 〈x, d〉 ∈ B or 〈x, d〉 ∈ C. Suppose 〈x, d〉 ∈ C. Let y be theunique witness such that the e-th Turing program with input x yieldsan output in < y steps. Then there exists z ≤ y such that the d-thTuring program with input 〈x, d〉 yields an output in < z steps. Then〈x, d〉 ∈ D, which contradicts thatD∩C = ∅. Thus we have 〈x, d〉 ∈ B.

Let y be the unique witness such that the e-th Turing program withinput x yields an output in < y steps. Since 〈x, d〉 ∈ D, we have forsome z, the d-th Turing program with input 〈x, d〉 yields an output in< z steps. Since for all z ≤ y, the d-th Turing program with input〈x, d〉 does not yield an output in < z steps, we have z > y. Thus, theleft-to-right direction holds.

Now we show that A ≤T D. If 〈x, d〉 /∈ D, then x /∈ A. If 〈x, d〉 ∈ D,from the above claim, we can effectively decide whether x ∈ A.

From Theorem 5.5, we can give a simpler proof of Theorem 3.1.

Proof. Suppose A is a non-recursive r.e. set. Pick the disjoint r.e.sets 〈B,C〉 as in Theorem 5.5. We show that 〈B,C〉 is a recursivelyinseparable pair, and B and C have the same Turing degree as A.

Suppose D is a recursive set which separates B and C. By Theorem5.5, A ≤T D and thus A is recursive which leads to a contradiction.

From Theorem 5.5, we have B,C ≤T A. Since B is a r.e. set whichseparates B and C, by Theorem 5.5, we have A ≤T B. Similarly, wehave A ≤T C. Thus, A, B and C have the same Turing degree.

Theorem 5.6 (Visser, Theorem 6, [18]). For any r.e. theory T withfinite signature, T is locally finitely satisfiable iff T is interpretable inR.13

13In fact, if T is locally finitely satisfiable, then T is interpretable in R via aone-piece one-dimensional parameter-free interpretation.

12 YONG CHENG

By Theorem 3.2, for any r.e. Turing degree 0 < d < 0′, there is anessentially undecidable theory with Turing degree d. We denote thistheory by Td.

Theorem 5.7. For any r.e. Turing degree 0 < d < 0′, the theory Tdis Turing persistent and Td R.

Proof. Suppose A is a r.e. set with Turing degree d, and Td is con-structed as in Theorem 3.2 via the recursively inseparable pair 〈B,C〉constructed as in Theorem 5.5.

Suppose V is a consistent r.e. extension of Td. Define D = n : V ⊢Φn. Note that B ⊆ D and D ∩ C = ∅. By Theorem 5.5, A ≤T D.Since Td has the same Turing degree as A, we have Td ≤T V . Thus,Td is Turing persistent.

Since Td is locally finitely satisfiable, we have TdR. If RTd, sinceR is Turing persistent, then R ≤T Td which leads to a contradiction.Thus Td R.

Theorem 5.8. For any non-recursive r.e. set A, we can uniformly finda recursively inseparable pair 〈B,C〉 such that:

(1) G1 holds for U〈B,C〉;(2) U〈B,C〉 R;(3) U〈B,C〉 has the same Turing degree as A;(4) U〈B,C〉 is Turing persistent.

Proof. Take the pair of r.e. sets 〈B,C〉 as in Theorem 5.5 such thatA,B,C have the same Turing degree. Since A is not recursive, 〈B,C〉is a recursively inseparable pair. From Theorem 5.5, we can uniformlyfind such a recursively inseparable pair 〈B,C〉 from a non-recursiver.e. set A. By Theorem 5.2, it suffices to prove (3) and (4).

(3): Since P(n) is in U〈B,C〉 iff n ∈ B, and ¬P(n) is in U〈B,C〉 iffn ∈ C, we have B and C are recursive in U〈B,C〉. By essentially thesame argument as Theorem 3.2, we can show that U〈B,C〉 is recursive in(B,C). The key point is that the theory U〈B,C〉 admits the eliminationof quantifiers. Thus, any sentence θ of the theory T is equivalent to adisjunctive normal form of 〈P(n) : n ∈ N〉, and this disjunctive normalform can be found explicitly once θ is explicitly given.

(4): Suppose V is a consistent r.e. extension of U〈B,C〉. Define D =n : V ⊢ P(n). Note that B ⊆ D and D ∩ C = ∅. By Theorem5.5, A ≤T D. Since U〈B,C〉 has the same Turing degree as A, we haveU〈B,C〉 ≤T V . Thus, U〈B,C〉 is Turing persistent.

It is an open question in [1]: can we show that for any Turing degree0 < d ≤ 0′, there is a theory U such that G1 holds for U , U R andU has Turing degree d? As a corollary of Theorem 5.8, the followingtheorem answers this question positively and proves a stronger result.

FINDING THE LIMIT OF INCOMPLETENESS II 13

Theorem 5.9. For any r.e. Turing degree 0 < d ≤ 0′, we can uni-formly find a Turing persistent theory T with Turing degree d such thatG1 holds for T and T R.

Since there are only countably many r.e. degrees, we have countablymany Turing persistent theories in D.

By default, theories refer to first order theories. Now, we show thatTheorem 5.9 also holds for theories in propositional logic. We workin propositional logic with countable many variables pn : n ∈ ω. Atheory in propositional logic is just a set of formulas in the language ofpropositional logic. We could view the language of propositional logicas a special instance of first order language: propositional variablescan be viewed as constants in first order language. Thus, our notion ofinterpretation also applies to theories in propositional logic.Theorem 5.10 (Jerabek, Theorem 4.5, [7]).

(1) For Σ02-axiomatized theory T , T is interpretable in some consistent

existential theory iff T is weakly interpretable in ECL for somelanguage L.14

(2) The theory R is not weakly interpretable in ECL for any languageL.

By Theorem 5.10, R is not interpretable in any consistent existentialtheory.

Theorem 5.11. For any r.e. Turing degree 0 < d ≤ 0′, there existsa Turing persistent theory Ud in proposition logic with Turing degree d

such that Ud R and G1 holds for Ud.

Proof. Suppose A is a r.e. set with Turing degree d, and 〈X, Y 〉 isthe recursively inseparable pair with Turing degree d constructed as inTheorem 5.5. Let Ud = pn : n ∈ X ∪ ¬pn : n ∈ Y . Note that Ud

is consistent.

Claim. Ud is essentially incomplete.

Proof. Let S be a recursively axiomatized consistent extension of Ud.Define B = n ∈ ω : S ⊢ pn and C = n ∈ ω : S ⊢ ¬pn. Note thatB,C are r.e. sets, X ⊆ B, and Y ⊆ C. Since 〈X, Y 〉 is a recursivelyinseparable pair, we have B ∪ C 6= ω. Thus, there exists n ∈ ω suchthat S 0 pn and S 0 ¬pn. Hence, S is incomplete.

Now we define a theory T as follows. The language of T consistsof the signature of R and an extra unary predicate symbol P . Theaxioms of T consist of axioms of R plus the following axioms: P (n) ifn ∈ X ; and ¬P (n) if n ∈ Y . We can show that Ud T by mappingpn to P (n). Since T is locally finitely satisfiable with finite signature,

14The theory of existentially closed L-structures (ECL) is the model completionof the empty theory in the language L (see [7]).

14 YONG CHENG

we have T R. Thus, Ud R. Since R is not interpretable in anyconsistent existential theory, we have Ud R.

Now we show that Ud has Turing degree d. Clearly, X, Y ≤T Ud.By the normal form theorem in propositional logic, any formula inproposition logic is equivalent to a disjunctive normal form. Thus,Ud ≤T (X, Y ). So the theory Ud has Turing degree d.

Note that for Turing persistent theories, if they are comparable w.r.t.interpretation, then they are comparable w.r.t. Turing degree. FromTheorem 5.9, given incomparable r.e. sets w.r.t. Turing degree, wecan find incomparable r.e. theories in D w.r.t. interpretation. It is anopen question in [1]: are elements of 〈D,〉 comparable? The follow-ing theorem answers this question negatively and provides much moreinformation.

Theorem 5.12. Given r.e. sets A <T B, there is a sequence of Turingpersistent r.e. theories 〈Sn : n ∈ ω〉 such that Sn ∈ D, A <T Sn <T Band Sn are incomparable w.r.t. interpretation.

Proof. By Fact 3.6, there exists a sequence of r.e. sets 〈Cn : n ∈ ω〉such that A <T Cn <T B and Cn are incomparable w.r.t. Turingdegreee. By Theorem 5.9, for each n, we can find a Turing persistentr.e. theory Sn ∈ D with the same Turing degree as Cn. Thus, A <T

Sn <T B for each n. Since each Sn is Turing persistent, and Cn’sare incomparable w.r.t. Turing degree, we have Sn’s are incomparablew.r.t. interpretation.

Lemma 5.13. For r.e. theories A and B, if G1 holds for both A andB, then G1 also holds for A⊕ B.

Proof. It suffices to show that A ⊕ B is essentially undecidable. Sup-pose U is a consistent decidable extension of A ⊕ B. Define X =〈pφq, pψq〉 : U ⊢ P → φ or U ⊢ ¬P → ψ. Since U is decidable,X is recursive. Note that A ⊆ (X)0 and B ⊆ (X)1. We claim thatat least one of (X)0 and (X)1 is consistent. If both (X)0 and (X)1are inconsistent, then U ⊢ (P →⊥) and U ⊢ (¬P →⊥). Thus, U ⊢⊥which contradicts that U is consistent. WLOG, we assume that (X)0 isconsistent. Then (X)0 is a consistent decidable extension of A, whichcontradicts that A is essentially undecidable.

As a corollary, the interpretation degree structure of r.e. theoriesfor which G1 holds with the operators ⊕ and ⊗ is also a distributivelattice.

From the email communication, Albert Visser shows that there is astrictly descending chain of essentially undecidable theories w.r.t. in-terpretation. The following theorem is inspired by Visser’s this result.

Theorem 5.14. Given r.e. sets A <T B, there is a sequence ofr.e. theories 〈Cn : n ∈ ω〉 such that Cn ∈ D, A ≤T Cn ≤T B, and

FINDING THE LIMIT OF INCOMPLETENESS II 15

〈. . . Cn+1 Cn . . .C0〉 is a strictly descending chain of elements ofD.

Proof. By Theorem 5.12, we can pick a sequence of Turing persistentr.e. theories 〈Sn : n ∈ ω〉 such that Sn ∈ D, A <T Sn <T B and Sn

are incomparable w.r.t. interpretation. Define Cn = Sn ⊕ . . .⊕ S0. ByLemma 5.13, Cn ∈ D. Note that A ≤T Cn ≤T B.

Now we show that Cn+1 Cn for any n ∈ ω. We prove by inductionon n. It is easy to show that C1C0 since S0 and S1 are incomparablew.r.t. interpretation. Now, we suppose Cn+1Cn and show that Cn+2

Cn+1. It suffices to show that Cn+1 5 Cn+2. Suppose not, i.e. Cn+1

Cn+2. Then Cn+1 Sn+2 ⊕ Cn+1 Sn+2.

Claim. Suppose T is a consistent r.e. theory. For any n, if Cn T ,then Si ≤T T for some 0 ≤ i ≤ n.

Proof. We prove by induction on n. If n = 0, the conclusion holdssince S0 is Turing persistent. Suppose Cn+1 = Sn+1 ⊕ Cn T , τ isthe interpretation of Cn+1 in T , and P is the new predicate used inSn+1 ⊕ Tn. If T + Pτ is consistent, then Sn+1 T + Pτ . Since Sn+1

is Turing persistent, Sn+1 ≤T T + Pτ ≤T T . Otherwise, T + ¬Pτ isconsistent. Then Cn T + ¬Pτ . By induction, Si ≤T T + ¬Pτ ≤T Tfor some 0 ≤ i ≤ n. Thus, we have Si ≤T T for some 0 ≤ i ≤ n+1.

By the above claim, we have Si ≤T Sn+2 for some 0 ≤ i ≤ n + 1,which leads to a contradiction. Thus, 〈. . . Cn+1 Cn . . . C0〉 is astrictly descending chain of elements of D.

As a corollary of Theorem 5.14, there are many strictly descendingchains of elements of D. The following theorem shows that the inter-pretation degree structure of r.e. theories for which G1 holds has nomaximal element.

Theorem 5.15. For any r.e. theory A for which G1 holds, we canuniformly find a r.e. theory B for which G1 holds such that AB.

Proof. Let A be any r.e. theory for which G1 holds. By Fact 3.6, wecan uniformly find a r.e. set C such that A is incomparable with Cw.r.t. Turing degree. By Theorem 5.9, from C we can uniformly finda Turing persistent theory T for which G1 holds such that T has thesame Turing degree as C. Let B = A ⊗ T . Suppose B A. SinceT B A and T is Turing persistent, T is Turing comparable with Awhich leads to a contradiction. Thus, AB.

Theorem 5.16. If 〈D,〉 has a minimal element, then it is also aminimum, and is not Turing persistent.

Proof. Suppose A is a minimal element of 〈D,〉. We show that forany r.e. theory B ∈ D, we have A B. Since A,B ∈ D, by Lemma

16 YONG CHENG

5.13, A ⊕ B ∈ D. Since A is minimal, we have A ⊕ B is mutuallyinterpretable with A. Thus, A B.

Now we show that A is not Turing persistent. Suppose A is Turingpersistent and d0 is the Turing degree of A. Take a r.e. Turing degree0 < d1 < 0′ such that d0 is incomparable with d1. By Theorem5.9, take a Turing persistent theory T ∈ D with the Turing degree d1.Since A is the minimum element of 〈D,〉, we have A T . Since A isTuring persistent, we have A ≤T T , which contradicts the fact that d0

is incomparable with d1.

By Theorem 5.14, we can effectively find many strictly descendingchains of elements of D. But it is unknown that whether for any r.e. the-ory A for which G1 holds, we can effectively find a r.e. theory B forwhich G1 holds such that BA. But for finite axiomatized theories, wecan effectively find such a theory B. We first introduce some notions.Definition 5.17 (The theory TN, [19]). The theory TN consists ofthe following axioms:

• ∀x(x ≮ 0); ∀x∀y∀z((x < y ∧ y < z) → x < z);• ∀x∀y∀z(x < y ∨ x = y ∨ y < x);• ∀x(Sx ≮ x);• ∀x∀y(x < y → (x < Sx ∧ y ≮ Sx));• ∀x(x + 0 = x);• ∀x∀y(x+ Sy = S(x+ y));• ∀x(x × 0 = 0);• ∀x∀y(x× Sy = x× y + x).

Note that a model of TN is a linear ordering that either representsa finite ordinal or starts with a copy of ω.

For Σ01-sentence ψ = ∃xφ(x) where φ is ∆0

0-sentence, define thefinitely axiomatizable theory [ψ] as follows:

[ψ] = TN+ ∃x∃y < xφ(y).

Definition 5.18 ([19]). Suppose ϕ = ∃xA(x) and ψ = ∃xB(x) aretwo Σ0

1-sentences. We Define:

(1) ϕ ψ , ∃x(A(x) ∧ ∀y < x¬B(y));(2) ϕ ≺ ψ , ∃x(B(x) ∧ ∀y ≤ x¬A(y));(3) If θ is ϕ ψ, then θ⊥ = ψ ≺ ϕ;(4) If θ is ϕ ≺ ψ, then θ⊥ = ψ ϕ.Fact 5.19 (Visser, [19]). Suppose ϕ, ψ are Σ0

1-sentences.

• If ψ is false, then [ψ] ⊇ R.• If ψ is true, then [ψ]⊤.• If ϕ ψ, then [ψ] ⊢ ϕ.• Let A = ϕ ψ. If ϕ (or ψ) holds, then either A holds or A⊥

holds.

FINDING THE LIMIT OF INCOMPLETENESS II 17

Theorem 5.20 (Harvey Friedman). For any finitely axiomatized the-ory A, if ⊤A, then there exists a finitely axiomatized theory B suchthat ⊤B A.15

Proof. By the fix point lemma, we can find a sentence θ such thatPA ⊢ θ ↔ (A A⊕ [θ]) (A⊕ [θ]⊤).

Claim. A 5 A⊕ [θ].

Proof. Suppose A A ⊕ [θ] holds. By Fact 5.19, either θ holds or θ⊥

holds.Case one: Suppose θ holds. By Fact 5.19, [θ] ⊤. Since A A ⊕

[θ] [θ], we have A⊤, which leads to a contradiction.Case two: Suppose ¬θ holds. Then θ⊥ holds. Thus, θ⊥ θ holds.

By Fact 5.19, [θ] ⊢ θ⊥ and thus [θ] ⊢ ⊥. Since ¬θ holds, we haveA⊕ [θ]⊤. Thus, A⊤, which leads to a contradiction.

By the similar argument, we can show that A ⊕ [θ] 5 ⊤. Thus,⊤A⊕ [θ]A.

Corollary 5.21. If A is a finitely axiomatized theory for which G1

holds and ⊤ A, then we can effectively find a finitely axiomatizedtheory B for which G1 holds such that ⊤ B A

Proof. This follows from Theorem 5.20. Take the finitely axiomatizedtheory B = A⊕ [θ] as in Theorem 5.20. Recall that the proof of Theo-rem 5.20 uses the fixed point lemma, but we can give an effective proofof the fixed point lemma. If θ is true, then [θ] ⊤ which contradictsthat BA. Thus, θ is false. Since [θ] ⊇ R, G1 holds for [θ]. By Lemma5.13, G1 holds for B.

From Corollary 5.21, the interpretation degree structure of finitelyaxiomatized theories for which G1 holds has no minimal element. Thefollowing theorem shows that there is no finitely axiomatized theoryinterpretable in R for which G1 holds.

Theorem 5.22. If T ∈ D, then T is not finitely axiomatized: i.e.,Dfinite ∩ D = ∅.

Proof. Suppose S ∈ Dfinite ∩ D. Since S R, S is locally finitelysatisfiable. Since S is finitely axiomatized, then S has a finite model,which contradicts the fact that S is essentially undecidable.

Moreover, for any Turing persistent r.e. theory A for which G1 holds,we can effectively find a weaker r.e. theory B than A w.r.t. interpre-tation such that G1 holds for B.

15This proof is simple than Friedman’s proof in [4], and the idea of this proof isdue to Visser in [19].

18 YONG CHENG

Theorem 5.23. For any Turing persistent r.e. theory A for which G1

holds, we can effectively find a r.e. theory B for which G1 holds suchthat B A.

Proof. Let A be any Turing persistent r.e. theory for which G1 holds.By Fact 3.6, we can effectively find a r.e. set C such that A is incom-parable with C w.r.t. Turing degree. By Theorem 5.9, from C we caneffectively find a Turing persistent theory T for which G1 holds suchthat T has the same Turing degree as C. Let B = A ⊕ T . We showthat B A. Suppose AB. Then AB T . Since A is Turing per-sistent, we have A ≤T C which contradicts the fact that A is Turingincomparable with C. Thus, B A.

Recall that we assume by default that the signature of the languageis finite. Finally, we show that whether 〈D,〉 has a minimal element(or 〈D,〉 is well founded) depends on the signature of the language.If the signature of the language is infinite, then 〈D,〉 has minimalelements.

In the following, we assume that the signature of theories is infinite.Now, we show that for any recursively inseparable pair 〈X, Y 〉, thereis a minimum theory T〈X,Y 〉 w.r.t. interpretation for which G1 holds.

Theorem 5.24. For any recursively inseparable pair 〈X, Y 〉, there isa theory T〈X,Y 〉 with infinite signature such that G1 holds for T〈X,Y 〉 andT〈X,Y 〉 is interpretable in any first order theory.

Proof. Let 〈X, Y 〉 be a recursively inseparable pair. Define the theoryT〈X,Y 〉 as follows. The language of T〈X,Y 〉 consists of a countable list ofunary predicate symbols 〈Pn : n ∈ ω〉. The axioms of T〈X,Y 〉 consist ofthe following:

(1) ∀xPn(x) if n ∈ X ;(2) ∃x¬Pn(x) if n ∈ Y .

Lemma 5.25. The theory T〈X,Y 〉 is essentially incomplete.

Proof. Let S be a recursively axiomatized consistent extension of T〈X,Y 〉.Define A = n ∈ ω : S ⊢ ∀xPn(x) and B = n ∈ ω : S ⊢ ∃x¬Pn(x).Note that A,B are r.e. sets, X ⊆ A, and Y ⊆ B. Since 〈X, Y 〉 is arecursively inseparable pair, we have A ∪ B 6= ω. Thus, there existsn ∈ ω such that n /∈ A ∪ B. Hence, S is incomplete.

Lemma 5.26. The theory T〈X,Y 〉 is interpretable in any first ordertheory.

Proof. For any n ∈ X , we interpret Pn(x) as x = x; and for any n ∈ Y ,we interpret Pn(x) as x 6= x.

FINDING THE LIMIT OF INCOMPLETENESS II 19

Theorem 5.24 shows that, interpretation for theories with infinitesignature is not a good notion for comparing essentially undecidabletheories since an essentially undecidable theory may be interpretablein a decidable theory.

6. Conclusion

Finally, we give some concluding remarks. The existence of non-recursive r.e. set is essential to understand the incompleteness phenom-enon. Given any non-recursive r.e. set A, we can uniformly constructa r.e. theory for which G1 holds with the same Turing degree as A. In[5], Harvey Friedman proves G2 for theories interpreting IΣ1 based onthe existence of a remarkable set which is equivalent to the existenceof non-recursive r.e. set.

We have shown that whether there is a minimal r.e. theory for whichG1 holds depends on the definition of minimality. The fact that there isno minimal r.e. theory and no maximal r.e. theory for which G1 holdsw.r.t. some degree structures shows that incompleteness is omnipresentand there is no limit of the first incompleteness theorem.

Both [1] and this paper are about the limit of G1. A natural questionis: what is the limit of the second incompleteness theorem (G2) ifany? Both mathematically and philosophically, G2 is more problematicthan G1. In the case of G1, we are mainly interested in the fact thatsome sentence is independent of the base theory. But in the case ofG2, we are also interested in the content of the consistency statement.We can say that G1 is extensional in the sense that we can constructa concrete independent mathematical statement without referring toarithmetization and provability predicate. However, G2 is intensionaland “whether the consistency of T is provable in T” depends on manyfactors such as the way of formalization, the base theory we use, theway of coding, the way to express consistency, the provability predicatewe use, the way we enumerate axioms of the base theory, etc. For thediscussion of the intensionality of G2, we refer to [2].

We assume that Con(T ) is the canonical arithmetic formula express-ing the consistency of the base theory T defined as ¬PrT (0 6= 0) whereboth the coding and provability predicate it uses are standard.16 Wedefine that G2 holds for a r.e. theory T if for any r.e. theory S inter-preting T , we have S 0 Con(S). Pavel Pudlak shows that for any r.e.theory T interpreting Q, T 0 Con(T ) (see [12]). Thus, G2 holds forQ. But from [11], G2 does not hold for R since we can find a theorymutually interpretable with R but it proves its consistency.

The following are two natural questions about the limit of G2 worthyof future examination.

16This means that the coding we use is the standard Godel coding, and the prov-ability predicate we use satisfies the Hilbert-Bernays-Lob derivability conditions.

20 YONG CHENG

Question 6.1.

(1) Is there a r.e. theory T such that G2 holds for T and T has Turingdegree less than 0′?

(2) Is there a r.e. theory T such that G2 holds for T and T Q?

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